Deck 61: Ellipses

Question 1

Select the graph for following equation.​ $$\frac { x ^ { 2 } } { 9 } + y ^ { 2 } = 1$$ ​

A)​
B)​
C)​
D)​
E)​

Answer 1

​

Question 2

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. ​
Vertices: $( \pm 7,0 )$ ;Foci: $( \pm 5,0 )$ ​

A) $\frac { x ^ { 2 } } { 49 } - \frac { y ^ { 2 } } { 24 } = 1$
B) $\frac { x ^ { 2 } } { 7 } + \frac { y ^ { 2 } } { 5 } = 1$
C) $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 24 } = 1$
D) $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 25 } = 1$
E) $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 24 } = 1$

Answer 2

$\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 24 } = 1$

Question 3

Find the standard form of the equation of the ellipse with the given characteristics. ​
Foci: $( 0,0 ) , ( 0,6 )$ ;major axis of length 12.
​

A) $\frac { x ^ { 2 } } { 27 } - \frac { ( y - 3 ) ^ { 2 } } { 36 } = 1$
B) $\frac { x ^ { 2 } } { 36 } + \frac { ( y - 3 ) ^ { 2 } } { 9 } = 1$
C) $\frac { x ^ { 2 } } { 27 } + \frac { ( y - 3 ) ^ { 2 } } { 36 } = 1$
D) $\frac { x ^ { 2 } } { 9 } + \frac { ( y + 3 ) ^ { 2 } } { 36 } = 1$
E) $\frac { x ^ { 2 } } { 27 } + \frac { y ^ { 2 } } { 36 } = 1$

Answer 3

$\frac { x ^ { 2 } } { 27 } + \frac { ( y - 3 ) ^ { 2 } } { 36 } = 1$

Question 4

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. ​
Vertices: $( 0 , \pm 8 )$ ;Foci: $( 0 , \pm 4 )$ ​

A) $\frac { x ^ { 2 } } { 48 } + \frac { y ^ { 2 } } { 64 } = 1$
B) $\frac { x ^ { 2 } } { 8 } + \frac { y ^ { 2 } } { 4 } = 1$
C) $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 16 } = 1$
D) $\frac { x ^ { 2 } } { 64 } - \frac { y ^ { 2 } } { 48 } = 1$
E) $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 48 } = 1$

Question 5

Select the graph for following equation.​ $$\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } = 1$$ ​

A)​
B)​
C)​
D)​
E)​

Question 6

Identify the conic as a circle or an ellipse then find the center.​ $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$ ​

A)Circle Center: (0,0)
B)Ellipse Center: (0,0)
C)​Circle Center: (3,2)
D)​Ellipse Center: (3,2)
E)​Circle Center: (2,3)

Question 7

Find the Foci of the conic.​ $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 64 } = 1$ ​

A)Foci: $( 0 , \pm \sqrt { 48 } )$
B)Foci: $( \pm \sqrt { 8 } , 0 )$
C)Foci: $( 0 , \pm \sqrt { 8 } )$
D)Foci: $( \pm \sqrt { 48 } , 0 )$
E)Foci: $( \sqrt { 8 } , \sqrt { 48 } )$

Question 8

Find the vertices of the conic.​ $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 36 } = 1$ ​

A)Vertices: $( 0 , \pm 3 )$
B)Vertices: $( \pm 6,0 )$
C)Vertices: $( 0 , \pm 6 )$
D)Vertices: $( \pm 3,6 )$
E)Vertices: $( \pm 3,0 )$

Question 9

Find the standard form of the equation of the ellipse with the given characteristics. ​
Vertices: $( 0,4 ) , ( 8,4 )$ ;minor axis of length 10.
​

A) $\frac { ( x + 4 ) ^ { 2 } } { 16 } + \frac { ( y + 4 ) ^ { 2 } } { 25 } = 1$
B) $\frac { ( x - 4 ) ^ { 2 } } { 16 } + \frac { ( y - 4 ) ^ { 2 } } { 25 } = 1$
C) $\frac { ( x + 4 ) ^ { 2 } } { 16 } + \frac { ( y - 4 ) ^ { 2 } } { 25 } = 1$
D) $\frac { ( x - 4 ) ^ { 2 } } { 16 } + \frac { ( y + 4 ) ^ { 2 } } { 25 } = 1$
E) $\frac { ( x - 4 ) ^ { 2 } } { 16 } - \frac { ( y - 4 ) ^ { 2 } } { 25 } = 1$

Question 10

Find the standard form of the equation of the ellipse with the given characteristics. ​
Center: $( 15,5 )$ ;a = 3b;Foci: $( 5,5 ) , ( 25,5 )$ ,where a - thr semimajor axis,b - semiminor axis.
​

A) $\frac { ( x - 5 ) ^ { 2 } } { 225 } + \frac { ( y - 5 ) ^ { 2 } } { 200 } = 1$
B) $\frac { x ^ { 2 } } { 225 } - \frac { ( y - 5 ) ^ { 2 } } { 200 } = 1$
C) $\frac { ( x - 5 ) ^ { 2 } } { 225 } + \frac { y ^ { 2 } } { 200 } = 1$
D) $\frac { ( x - 5 ) ^ { 2 } } { 225 } + \frac { ( y + 5 ) ^ { 2 } } { 200 } = 1$
E) $\frac { ( x + 5 ) ^ { 2 } } { 225 } + \frac { ( y + 5 ) ^ { 2 } } { 200 } = 1$

Question 11

Select the graph for following equation.​ $$\frac { ( x + 2 ) ^ { 2 } } { 9 } + \frac { ( y + 2 ) ^ { 2 } } { 4 } = 1$$ ​

A)​
B)​
C)​
D)​
E)​

Question 12

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. ​
Vertical major axis passes through the points $( 0,7 ) \text { and } ( 5,0 )$ .
​

A) $\frac { x ^ { 2 } } { 7 } + \frac { y ^ { 2 } } { 5 } = 1$
B) $\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 49 } = 1$
C) $\frac { x ^ { 2 } } { 49 } - \frac { y ^ { 2 } } { 25 } = 1$
D) $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 49 } = 1$
E) $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 25 } = 1$

Question 13

Find the standard form of the equation of the ellipse with the given characteristics. ​
Foci: $( 0,0 ) , ( 6,0 )$ ;major axis of length 10.
​

A) $\frac { ( x - 3 ) ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 25 } = 1$
B) $\frac { ( x - 3 ) ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 25 } = 1$
C) $\frac { ( x + 3 ) ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 25 } = 1$
D) $\frac { ( x - 3 ) ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } = 1$
E) $\frac { ( x + 3 ) ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 16 } = 1$

Question 14

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. ​
Foci: $( \pm 5,0 )$ ;major axis of length 12
​

A) $\frac { x ^ { 2 } } { 36 } - \frac { y ^ { 2 } } { 11 } = 1$
B) $\frac { x ^ { 2 } } { 6 } + \frac { y ^ { 2 } } { 5 } = 1$
C) $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 25 } = 1$
D) $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 11 } = 1$
E) $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 11 } = 1$

Question 15

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. ​
Foci: $( \pm 2,0 )$ ;major axis of length 18
​

A) $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 77 } = 1$
B) $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 2 } = 1$
C) $\frac { x ^ { 2 } } { 81 } - \frac { y ^ { 2 } } { 77 } = 1$
D) $\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 4 } = 1$
E) $\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 77 } = 1$

Question 16

Find the standard form of the equation of the ellipse with the given characteristics. ​
Center: $( 0,4 )$ ;a = 2c;Vertices: $( - 8,8 ) , ( 8,8 )$ ,where a - thr semimajor axis,b - semiminor axis and $c ^ { 2 } = a ^ { 2 } - b ^ { 2 }$ .
​

A) $\frac { ( x - 8 ) ^ { 2 } } { 64 } + \frac { ( y - 8 ) ^ { 2 } } { 48 } = 1$
B) $\frac { x ^ { 2 } } { 64 } - \frac { ( y - 8 ) ^ { 2 } } { 48 } = 1$
C) $\frac { x ^ { 2 } } { 64 } - \frac { ( y + 8 ) ^ { 2 } } { 48 } = 1$
D) $\frac { x ^ { 2 } } { 64 } + \frac { ( y - 8 ) ^ { 2 } } { 48 } = 1$
E) $\frac { ( x - 8 ) ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 48 } = 1$

Question 17

Select the graph for following equation.​ $$\frac { ( x - 2 ) ^ { 2 } } { 16 } + ( y + 1 ) ^ { 2 } = 1$$ ​

A)​
B)​
C)​
D)​
E)​

Question 18

Select the graph for following equation.​ $$\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 16 } = 1$$ ​

A)​
B)​
C)​
D)​
E)​

Question 19

Find the Eccentricity of the conic.​ $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 16 } = 1$ ​

A)Eccentricity: $e = \frac { \sqrt { 48 } } { 64 }$
B)Eccentricity: $e = \frac { \sqrt { 48 } } { 4 }$
C)Eccentricity: $e = \frac { \sqrt { 64 } } { 4 }$
D)Eccentricity: $e = \frac { \sqrt { 48 } } { 8 }$
E)Eccentricity: $e = \frac { \sqrt { 48 } } { 48 }$

Question 20

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin.​ ​

A) $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 4 } = 1$
B) $\frac { x } { 4 } + \frac { y } { 16 } = 1$
C) $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 16 } = 1$
D) $\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 16 } = 1$
E) $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 16 } = 1$

Question 21

The first artificial satellite to orbit Earth was Sputnik I (launched by the former Soviet Union in 1957).Its highest point above Earths surface was 943 kilometers,and its lowest point was 211 kilometers (see figure).The center of Earth was at one focus of the elliptical orbit,and the radius of Earth is 6392 kilometers.Find the eccentricity of the orbit.Round your results to four decimal place.​ ​ $$x = 211 \mathrm {~km} , y = 943 \mathrm {~km}$$ ​

A)Eccentricity: $e \approx 0.4555$
B)Eccentricity: $e \approx 0.4755$
C)Eccentricity: $e \approx 0.4955$
D)Eccentricity: $e \approx 0.4355$
E)Eccentricity: $e \approx 0.5155$

Question 22

Find the standard form of the equation of the ellipse with the given characteristics. ​
Center: $( - 6,4 )$ ,a = 5c,foci: $( - 8,4 ) , ( - 4,4 )$ ,where a - thr semimajor axis,b - semiminor axis and $c ^ { 2 } = a ^ { 2 } - b ^ { 2 }$ .

A) $\frac { ( x - 6 ) ^ { 2 } } { 96 } + \frac { ( y + 4 ) ^ { 2 } } { 100 } = 1$
B) $\frac { ( x + 6 ) ^ { 2 } } { 96 } + \frac { ( y - 4 ) ^ { 2 } } { 100 } = 1$
C) $\frac { ( x + 6 ) ^ { 2 } } { 100 } + \frac { ( y - 4 ) ^ { 2 } } { 96 } = 1$
D) $\frac { ( x - 4 ) ^ { 2 } } { 100 } + \frac { ( y + 6 ) ^ { 2 } } { 96 } = 1$
E) $\frac { ( x + 4 ) ^ { 2 } } { 100 } + \frac { ( y - 6 ) ^ { 2 } } { 96 } = 1$

Question 23

Find the standard form of the equation of the ellipse with the following characteristics. ​
Foci: $( \pm 3,0 )$ ,major axis of length: 16
​

A) $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 55 } = 1$
B) $\frac { x ^ { 2 } } { 256 } + \frac { y ^ { 2 } } { 247 } = 1$
C) $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 9 } = 1$
D) $\frac { x ^ { 2 } } { 256 } + \frac { y ^ { 2 } } { 9 } = 1$
E) $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 64 } = 1$

Question 24

Find the center and radius of the conic.​ $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 64 } = 1$ ​

A)Center: $( 0,8 )$ Radius: 8
B)Center: $( 0,0 )$ Radius: 8
C)Center: (0,0) Radius: $64$
D)Center: $( 8,0 )$ Radius: 8
E)Center: $( 0,8 )$ Radius: $64$

Question 25

Find the standard form of the equation of the ellipse with the following characteristics. ​
Foci: $( \pm 2,0 )$ ,major axis of length: 22
​
​

A) $\frac { x ^ { 2 } } { 121 } + \frac { y ^ { 2 } } { 117 } = 1$
B) $\frac { x ^ { 2 } } { 484 } + \frac { y ^ { 2 } } { 4 } = 1$
C) $\frac { x ^ { 2 } } { 121 } + \frac { y ^ { 2 } } { 4 } = 1$
D) $\frac { x ^ { 2 } } { 484 } + \frac { y ^ { 2 } } { 480 } = 1$
E) $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 121 } = 1$

Question 26

A semielliptical arch over a tunnel for a one-way road through a mountain has a major axis of 54 feet and a height at the center of 12 feet.Find an equation of the semielliptical arch. ​

A) $\frac { x ^ { 2 } } { 144 } - \frac { y ^ { 2 } } { 729 } = 1$
B) $\frac { x ^ { 2 } } { 729 } - \frac { y ^ { 2 } } { 144 } = 1$
C) $\frac { x ^ { 2 } } { 729 } + \frac { y ^ { 2 } } { 144 } = 1$
D) $\frac { x ^ { 2 } } { 54 } + \frac { y ^ { 2 } } { 729 } = 1$
E) $\frac { x ^ { 2 } } { 144 } + \frac { y ^ { 2 } } { 729 } = 1$

Question 27

Find the standard form of the equation of the ellipse with the given characteristics. ​
Foci: $( 5,3 ) , ( 5,13 )$ ,endpoints of the major axis: $( 5,0 ) , ( 5,16 )$ ​

A) $\frac { ( x + 8 ) ^ { 2 } } { 64 } + \frac { ( y - 5 ) ^ { 2 } } { 39 } = 1$
B) $\frac { ( x + 5 ) ^ { 2 } } { 64 } + \frac { ( y - 8 ) ^ { 2 } } { 39 } = 1$
C) $\frac { ( x - 5 ) ^ { 2 } } { 39 } + \frac { ( y + 8 ) ^ { 2 } } { 64 } = 1$
D) $\frac { ( x + 5 ) ^ { 2 } } { 39 } + \frac { ( y - 8 ) ^ { 2 } } { 64 } = 1$
E) $\frac { ( x - 8 ) ^ { 2 } } { 64 } + \frac { ( y + 5 ) ^ { 2 } } { 39 } = 1$

Question 28

Find the center and foci of the ellipse.​ $\frac { ( x + 1 ) ^ { 2 } } { 7 } + \frac { ( y - 5 ) ^ { 2 } } { 16 } = 1$ ​

A)center: $( 1 , - 5 )$ foci: $( 1,4 ) , ( 1 , - 2 )$
B)center: $( - 1,5 )$ foci: $( 8,5 ) , ( 2,5 )$
C)center: $( - 1,5 )$ foci: $( - 1,2 ) , ( - 1,8 )$
D)center: $( - 1,5 )$ foci: $( - 2 , - 5 ) , ( 4 , - 5 )$
E)center: $( 1 , - 5 )$ foci: $( - 2,5 ) , ( 4,5 )$

Question 29

An elliptical stained-glass insert is to be fitted in a 6ft by 4 ft rectangular opening (see figure).Using the coordinate system shown,find an equation for the ellipse.​ ​

A) $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$
B) $\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 9 } = 1$
C) $\frac { x ^ { 2 } } { 3 } - \frac { y ^ { 2 } } { 2 } = 1$
D) $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 4 } = 1$
E) $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 9 } = 1$

Question 30

Select the graph of the conic.​ $$\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 64 } = 1$$ ​

A)​
B)​
C)​ ​
D)​ ​
​
E)​

Question 31

Identify the conic as a circle or an ellipse then find the center.​ $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 49 } = 1$ ​

A)Circle Center: (0,0)
B)Ellipse Center: (0,0)
C)Circle Center: (7,0)
D)Ellipse Center: (0,7)
E)​Circle Center: (7,7)

Question 32

Find the standard form of the equation of the ellipse with the given characteristics. ​
​vertices: (-1,-11), (-1,7),minor axis of length 2
​

A) $\frac { ( x + 1 ) ^ { 2 } } { 81 } + \frac { ( y + 2 ) ^ { 2 } } { 4 } = 1$
B) $\frac { ( x + 2 ) ^ { 2 } } { 81 } + \frac { ( y + 1 ) ^ { 2 } } { 4 } = 1$
C) $\frac { ( x + 1 ) ^ { 2 } } { 4 } + \frac { ( y + 2 ) ^ { 2 } } { 81 } = 1$
D) $\frac { ( x - 2 ) ^ { 2 } } { 81 } + \frac { ( y - 1 ) ^ { 2 } } { 4 } = 1$
E) $\frac { ( x - 1 ) ^ { 2 } } { 4 } + \frac { ( y - 2 ) ^ { 2 } } { 81 } = 1$

Question 33

Find the eccentricity of the ellipse.​ $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 81 } = 1$ ​

A)Eccentricity: $e = \frac { 8 \sqrt { 17 } } { 17 }$
B)Eccentricity: $e = \frac { \sqrt { 17 } } { 9 }$
C)Eccentricity: $$e = - \frac { \sqrt { 17 } } { 9 }$$
D)Eccentricity: $$e = \frac { \sqrt { 17 } } { 17 }$$
E)Eccentricity: $e = - \frac { 9 \sqrt { 17 } } { 9 }$

Question 34

Identify the conic as a circle or an ellipse.​ $$x ^ { 2 } + y ^ { 2 } - 3 x + 4 y - 2 = 0$$ ​

A)Circle
B)Ellipse

Question 35

Find the center and vertices of the ellipse.​ $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 36 } = 1$ ​

A)center: (8,6)vertices: (-8,-6), (8,6)
B)center: (0,0)vertices: (0,-8), (0,8)
C)center: (0,0)vertices: (-8,0), (8,0)
D)center: (8,0)vertices: (0,-6), (0,6)
E)center: (0,0)vertices: (-6,0), (6,0)

Question 36

Match the equation with its graph. $$x ^ { 2 } + 9 y ^ { 2 } = 9$$

A)​
B)​
C)​
D)​
E)​

Question 37

Find the standard form of the equation of the ellipse with the given characteristics. ​
Foci: $( - 5 , - 12 ) , ( - 5 , - 4 )$ ,endpoints of the major axis: $( - 5 , - 16 ) , ( - 5,0 )$ ​

A) $\frac { ( x + 5 ) ^ { 2 } } { 64 } + \frac { ( y + 8 ) ^ { 2 } } { 48 } = 1$
B) $\frac { ( x - 5 ) ^ { 2 } } { 48 } + \frac { ( y - 8 ) ^ { 2 } } { 64 } = 1$
C) $\frac { ( x - 8 ) ^ { 2 } } { 64 } + \frac { ( y - 5 ) ^ { 2 } } { 48 } = 1$
D) $\frac { ( x + 8 ) ^ { 2 } } { 64 } + \frac { ( y + 5 ) ^ { 2 } } { 48 } = 1$
E) $\frac { ( x + 5 ) ^ { 2 } } { 48 } + \frac { ( y + 8 ) ^ { 2 } } { 64 } = 1$

Question 38

Find an equation of the ellipse with vertices $( \pm 9,0 )$ and eccentricity $e = \frac { 3 } { 9 }$ . ​

A) $\frac { x ^ { 2 } } { 72 } - \frac { y ^ { 2 } } { 81 } = 1$
B) $\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 72 } = 1$
C) $\frac { x ^ { 2 } } { 81 } - \frac { y ^ { 2 } } { 72 } = 1$
D) $\frac { x ^ { 2 } } { 72 } + \frac { y ^ { 2 } } { 81 } = 1$
E) $\frac { x ^ { 2 } } { 72 } + \frac { y ^ { 2 } } { 81 } = 1$ .

Question 39

Identify the conic by writing the equation in standard form.​ $18 y ^ { 2 } - 24 x ^ { 2 } + 216 y + 336 x - 543 = 0$ ​

A)​ $\frac { ( y - 6 ) ^ { 2 } } { \frac { 149 } { 5 } } + \frac { ( x - 7 ) ^ { 2 } } { \frac { 149 } { 10 } } = 1$ ;ellipse
B)​ $\frac { ( y + 6 ) ^ { 2 } } { \frac { 97 } { 6 } } - \frac { ( x - 7 ) ^ { 2 } } { \frac { 97 } { 8 } } = 1$ ;hyperbola
C)​ $\frac { ( y - 6 ) ^ { 2 } } { \frac { 5 } { 6 } } - \frac { ( x - 7 ) ^ { 2 } } { \frac { 5 } { 8 } } = 1$ ;hyperbola
D)​ $\frac { ( y + 6 ) ^ { 2 } } { \frac { 5 } { 6 } } - \frac { ( x - 7 ) ^ { 2 } } { \frac { 5 } { 8 } } = 1$ ;hyperbola
E)​ $\frac { ( y + 6 ) ^ { 2 } } { \frac { 149 } { 5 } } + \frac { ( x - 7 ) ^ { 2 } } { \frac { 149 } { 10 } } = 1$ ;ellipse

Question 40

Identify the conic as a circle or an ellipse then find the radius for circle or center for ellipse.​ $\frac { x ^ { 2 } } { 4 / 49 } + \frac { ( y + 1 ) ^ { 2 } } { 4 / 49 } = 1$ ​

A)Ellipse Center: $\frac { 2 } { 7 }$
B)Ellipse Center: 1
C)Circle Radius: 1
D)Circle Radius: $\frac { 2 } { 7 }$
E)None of the above

Question 41

Identify the graph of the following ellipse.​ $$\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 4 } = 1$$ ​

A) ​
B) ​
C) ​

Question 42

Find the center and vertices of the ellipse.​ $x ^ { 2 } + 9 y ^ { 2 } + 16 x + 18 y + 64 = 0$ ​

A)center: $( - 1 , - 8 )$ vertices: $( - 4 , - 8 ) , ( 9 , - 8 )$
B)center: $( 8,1 )$ vertices: $( 7,1 ) , ( 9,1 )$
C)center: $( - 8 , - 1 )$ vertices: $( - 9 , - 1 ) , ( - 7 , - 1 )$
D)center: $( - 8 , - 1 )$ vertices: $( - 11 , - 1 ) , ( - 5 , - 1 )$
E)center: $( 8,1 )$ vertices: $( 5,1 ) , ( - 11,1 )$

Question 43

Write the equation of the ellipse that has its center at the origin with focus at (0,1)and vertex at (0,11). ​

A) $\frac { x ^ { 2 } } { 121 } + \frac { y ^ { 2 } } { 120 } = 1$
B) $\frac { x ^ { 2 } } { 120 } - \frac { y ^ { 2 } } { 121 } = 1$
C) $\frac { x ^ { 2 } } { 120 } + \frac { y ^ { 2 } } { 121 } = - 1$
D) $\frac { x ^ { 2 } } { 120 } + \frac { y ^ { 2 } } { 121 } = 1$
E)none of these

Question 44

Write the equation of the ellipse that has its center at the origin,focus at (1,0)and vertex at (9,0). ​

A) $$\frac { x ^ { 2 } } { 80 } + \frac { y ^ { 2 } } { 81 } = 1$$
B) $\frac { x ^ { 2 } } { 1 } + \frac { y ^ { 2 } } { 9 } = 1$
C) $\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 80 } = - 1$
D) $\frac { x ^ { 2 } } { 81 } - \frac { y ^ { 2 } } { 80 } = 1$
E) $\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 80 } = 1$

Question 45

Find the standard form of the equation of the ellipse with vertices $( 0 , \pm 9 )$ and eccentricity $e = \frac { 3 } { 9 }$ .

A) $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 81 } = 1$
B) $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 81 } = 1$
C) $\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 9 } = 1$
D) $\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 72 } = 1$
E) $\frac { x ^ { 2 } } { 72 } + \frac { y ^ { 2 } } { 81 } = 1$

Question 46

Find the center and foci of the ellipse.​ $$\frac { ( x - 2 ) ^ { 2 } } { 45 } + \frac { ( y - 8 ) ^ { 2 } } { 81 } = 1$$ ​

A)center: $( - 2 , - 8 )$ ;foci: $( - 2 , - 14 ) , ( - 2 , - 2 )$
B)center: $( 2,8 )$ ;foci: $( 2,2 ) , ( 2,14 )$
C)center: $( 2,8 )$ ;foci: $( 14,8 ) , ( 8,8 )$
D)center: $( - 2 , - 8 )$ ;foci: $( - 8,8 ) , ( - 14,8 )$
E)center: $( 2,8 )$ ;foci: $( - 8 , - 8 ) , ( - 14 , - 8 )$

Question 47

Find the standard form of the equation of the ellipse with vertices $$( 0 , \pm 8 )$$ and eccentricity $e = \frac { 3 } { 8 }$ .

A) $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 64 } = 1$
B) $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 64 } = 1$
C) $\frac { x ^ { 2 } } { 55 } + \frac { y ^ { 2 } } { 64 } = 1$
D) $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 9 } = 1$
E) $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 55 } = 1$

Question 48

Find the center and vertices of the ellipse.​ $$x ^ { 2 } + 4 y ^ { 2 } - 18 x + 64 y + 333 = 0$$ ​

A)center: (-8,9),vertices: (-10,9), (-6,9)
B)center: (9,-8),vertices: (7,-8), (11,-8)
C)center: (-9,8),vertices: (-10,8), (-8,8)
D)center: (-9,8),vertices: (-11,8), (-7,8)
E)center: (9,-8),vertices: (8,-8), (10,-8)

Question 49

Find the standard form of the equation of the ellipse with vertices $( \pm 9,0 )$ and eccentricity $e = \frac { 7 } { 9 }$ .

A) $\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 49 } = 1$
B) $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 81 } = 1$
C) $\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 32 } = 1$
D) $\frac { x ^ { 2 } } { 32 } + \frac { y ^ { 2 } } { 81 } = 1$
E) $\frac { x ^ { 2 } } { 49 } - \frac { y ^ { 2 } } { 81 } = 1$

Question 50

Find the center and vertices of the ellipse.​ $x ^ { 2 } + 16 y ^ { 2 } - 10 x - 32 y + 25 = 0$ ​

A)center: $( 1,5 )$ ;vertices: $( - 3,5 ) , ( 5,5 )$
B)center: $( 5,1 )$ ;vertices: $( 1,1 ) , ( 9,1 )$
C)center: $( - 5 , - 1 )$ ;vertices: $( - 6 , - 1 ) , ( - 4 , - 1 )$
D)center: $( - 5 , - 1 )$ ;vertices: $( - 9 , - 1 ) , ( 1 , - 1 )$
E)center: $( 5,1 )$ ;vertices: $( 4,1 ) , ( 6,1 )$

Question 51

Identify the conic by writing the equation in standard form.​ $4 x ^ { 2 } + 9 y ^ { 2 } + 40 x + 54 y + 145 = 0$ ​

A) $\frac { ( x + 5 ) ^ { 2 } } { 9 } + \frac { ( y + 3 ) ^ { 2 } } { 4 } = 1$ ;ellipse
B) $( 3 x + 5 ) ^ { 2 } + ( 2 y + 1 ) ^ { 2 } = 167$ ;circle
C) $\frac { ( x + 5 ) ^ { 2 } } { \frac { 422 } { 9 } } + \frac { ( y + 3 ) ^ { 2 } } { \frac { 211 } { 2 } } = 1$ ;ellipse
D) $\frac { ( x + 5 ) ^ { 2 } } { \frac { 73 } { 3 } } + \frac { ( y + 3 ) ^ { 2 } } { \frac { 219 } { 4 } } = 1$ ;ellipse
E) $\frac { ( x + 5 ) ^ { 2 } } { \frac { 167 } { 9 } } + \frac { ( y + 3 ) ^ { 2 } } { \frac { 167 } { 4 } } = 1$ ;ellipse

Question 52

Sketch the graph of the ellipse,using the latera recta.​ $$\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 16 } = 1$$ ​

A)​
B)​
C)​
D)​
E)​

Question 53

Identify the conic by writing the equation in standard form.​ $4 x ^ { 2 } + 4 y ^ { 2 } + 8 x - 16 y - 12 = 0$ ​

A) $\frac { ( x + 1 ) ^ { 2 } } { \frac { 2 } { 4 } } + \frac { ( y - 2 ) ^ { 2 } } { \frac { 2 } { 4 } } = 8$ ;ellipse
B) $( x + 1 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 4$ ;circle
C) $( x + 1 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 8$ ;circle
D) $( x + 1 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 8$ ;circle
E) $\frac { ( x + 1 ) ^ { 2 } } { \frac { 17 } { 4 } } + \frac { ( y - 2 ) ^ { 2 } } { \frac { 17 } { 4 } } = 8$ ;ellipse

Question 54

Find the center and vertices of the ellipse.​ $$\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 36 } = 1$$ ​

A)center: (9,6)vertices: (-9,-6), (9,6)
B)center: (0,0)vertices: (0,-9), (0,9)
C)center: (9,0)vertices: (0,-6), (0,6)
D)center: (0,0)vertices: (-6,0), (6,0)
E)center: (0,0)vertices: (-9,0), (9,0)

Question 55

Sketch the graph of the ellipse,using the latera recta.​ $$9 x ^ { 2 } + 4 y ^ { 2 } = 36$$ ​

A)​
B)​
C)​
D)​
E)​

Question 56

An elliptical stained-glass insert is to be fitted in a 4ft by 6ft rectangular opening (see figure). Using the coordinate system shown,find an equation for the ellipse. ​

A) $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$
B) $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 9 } = 1$
C) $$\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 9 } = 1$$
D) $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 4 } = 1$
E) $\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 6 } = 1$

Question 57

Find the center and vertices of the ellipse.​ $x ^ { 2 } + 4 y ^ { 2 } - 6 x + 8 y + 9 = 0$ ​

A)center: (-1,3),vertices: (-3,3), (1,3)
B)center: (-3,1),vertices: (-4,1), (-2,1)
C)center: (3,-1),vertices: (2,-1), (4,-1)
D)center: (3,-1),vertices: (1,-1), (5,-1)
E)center: (-3,1),vertices: (-5,1), (-1,1)

Question 58

Find the center and foci of the ellipse.​ $$\frac { ( x - 7 ) ^ { 2 } } { 55 } + \frac { ( y - 2 ) ^ { 2 } } { 64 } = 1$$ ​

A)center: (-7,-2);foci: (-7,-5), (-7,1)
B)center: (7,2);foci: (-10,-2), (-4,-2)
C)center: (-7,-2);foci: (-10,2), (-4,2)
D)center: (7,2);foci: (7,-1), (7,5)
E)center: (7,2);foci: (4,2), (10,2)

Question 59

Find the graph of the following ellipse.​ $$4 x ^ { 2 } + 4 y ^ { 2 } - 40 x - 24 y + 120 = 0$$ ​

A)​
B)​
C)​
D)​ ​

Question 60

Find the center and the vertices of the ellipse.​ $81 x ^ { 2 } + y ^ { 2 } = 81$ ​

A)center: (-9,9);vertices: (-1,-9), (1,9)
B)center: (0,0);vertices: (0,-9), (0,9)
C)center: (0,0);vertices: (-9,-1), (9,1)
D)center: (0,0);vertices: (-9,0), (9,0)
E)center: (-9,9);vertices: (-1,0), (1,0)

Question 61

Find the standard form of the equation of the ellipse with the given characteristics. ​
Foci: $( 7 , - 7 ) , ( 7 , - 1 )$ ,endpoints of the major axis: $( 7 , - 13 ) , ( 7,5 )$ ​

A) $\frac { ( x + 7 ) ^ { 2 } } { 72 } + \frac { ( y - 4 ) ^ { 2 } } { 81 } = 1$
B) $\frac { ( x - 7 ) ^ { 2 } } { 72 } + \frac { ( y + 4 ) ^ { 2 } } { 81 } = 1$
C) $\frac { ( x - 7 ) ^ { 2 } } { 81 } + \frac { ( y + 4 ) ^ { 2 } } { 72 } = 1$
D) $\frac { ( x - 7 ) ^ { 2 } } { 72 } + \frac { ( y + 4 ) ^ { 2 } } { 81 } = - 1$
E) $\frac { ( x - 4 ) ^ { 2 } } { 81 } + \frac { ( y + 7 ) ^ { 2 } } { 72 } = 1$

Question 62

Find the standard form of the equation of the ellipse with the given characteristics. ​
Foci: $$( - 5 , - 1 ) , ( - 5,5 )$$ ,endpoints of the major axis: $$( - 5 , - 5 ) , ( - 5,9 )$$ ​

A) $\frac { ( x - 5 ) ^ { 2 } } { 40 } + \frac { ( y + 2 ) ^ { 2 } } { 49 } = 1$
B) $\frac { ( x + 5 ) ^ { 2 } } { 40 } + \frac { ( y - 2 ) ^ { 2 } } { 49 } = 1$
C) $\frac { ( x + 5 ) ^ { 2 } } { 49 } + \frac { ( y - 2 ) ^ { 2 } } { 40 } = 1$
D) $\frac { ( x - 2 ) ^ { 2 } } { 49 } + \frac { ( y + 5 ) ^ { 2 } } { 40 } = 1$
E) $\frac { ( x + 2 ) ^ { 2 } } { 49 } + \frac { ( y - 5 ) ^ { 2 } } { 40 } = 1$

Question 63

Sketch the graph of the ellipse,using the latera recta.​ ​

Question 64

Write the equation of the ellipse with foci at (2,2), (6,2),and b = 4,where b - semiminor axis. ​

A) $\frac { ( x - 4 ) ^ { 2 } } { 20 } - \frac { ( y - 2 ) ^ { 2 } } { 16 } = 1$
B) $\frac { ( x - 4 ) ^ { 2 } } { 20 } + \frac { ( y - 2 ) ^ { 2 } } { 16 } = 1$
C) $\frac { ( x - 4 ) ^ { 2 } } { 20 } + \frac { ( y - 2 ) ^ { 2 } } { 16 } = - 1$
D) $\frac { ( x + 4 ) ^ { 2 } } { 20 } + \frac { ( y + 2 ) ^ { 2 } } { 16 } = 1$
E) $\frac { ( x - 4 ) ^ { 2 } } { 16 } + \frac { ( y - 2 ) ^ { 2 } } { 20 } = 1$

Question 65

Find the standard form of the equation of the ellipse with the given characteristics. ​
Vertices: (6,1), (6,17),minor axis of length 3
​

A) $\frac { ( x + 6 ) ^ { 2 } } { 9 } + \frac { ( y + 9 ) ^ { 2 } } { 64 } = 1$
B) $\frac { ( x - 6 ) ^ { 2 } } { 9 } + \frac { ( y - 9 ) ^ { 2 } } { 64 } = 1$
C) $\frac { ( x - 6 ) ^ { 2 } } { 64 } + \frac { ( y - 9 ) ^ { 2 } } { 9 } = 1$
D) $\frac { ( x - 9 ) ^ { 2 } } { 64 } + \frac { ( y - 6 ) ^ { 2 } } { 9 } = 1$
E) $\frac { ( x + 9 ) ^ { 2 } } { 64 } + \frac { ( y + 6 ) ^ { 2 } } { 9 } = 1$

Question 66

Find the standard form of the equation of the ellipse with the following characteristics. ​
Foci: $( \pm 3,0 )$ ,major axis of length: 16
​

A) $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 55 } = 1$
B) $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 9 } = 1$
C) $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 64 } = 1$
D) $\frac { x ^ { 2 } } { 256 } + \frac { y ^ { 2 } } { 9 } = 1$
E) $\frac { x ^ { 2 } } { 256 } + \frac { y ^ { 2 } } { 247 } = 1$

Question 67

Find the standard form of the equation of the ellipse with the following characteristics. ​
Foci: $( \pm 5,0 )$ ,major axis of length: 18
​

A) $\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 56 } = 1$
B) $\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 25 } = 1$
C) $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 81 } = 1$
D) $\frac { x ^ { 2 } } { 324 } + \frac { y ^ { 2 } } { 25 } = 1$
E) $\frac { x ^ { 2 } } { 324 } + \frac { y ^ { 2 } } { 299 } = 1$

Question 68

Any sound from one focus of an ellipse reflects off the ellipse directly back to the other focus.This property explains whispering galleries such as Statuary Hall in Washington,D.C.The whispering gallery shown in illustration has the shape of a semi-circle.Find the distance sound travels as it leaves focus F and returns to focus F'.Round your answer to one decimal place,if necessary.​ ​

A)16 meters
B)16.8 meters
C)33.5 meters
D)32 meters
E)31.2 meters